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Volume 5, Number 2 (2013), 155–184
doi:10.7153/dea-05-11
HYBRID FIXED POINT THEORY IN PARTIALLY ORDERED
NORMED LINEAR SPACES AND APPLICATIONS
TO FRACTIONAL INTEGRAL EQUATIONS
BAPURAO C. D HAGE
(Communicated by Sotiris K. Ntouyas)
Abstract. In this paper, some basic hybrid fixed point theorems of Banach and Schauder type
and some hybrid fixed point theorems of Krasnoselskii type involving the sum of two operators
are proved in a partially ordered normed linear spaces which are further applied to nonlinear
Volterra fractional integral equations for proving the existence of solutions under certain monotonic conditions blending with the existence of either a lower or an upper solution type function.
This research is dedicated in the loving memory of my late father and mother who
imbibed in me the honesty, hard-work and services for all.
1. Introduction
It is well-known that the hybrid fixed point theorems which are obtained using the
mixed arguments from different branches of mathematics are very rich in applications
to allied areas of mathematics, particularly to the theory of nonlinear differential and
integral equations. See Aman [1], Heikkil̈a and Lakshmikantham [13], Zeidler [22]
and the references given therein. Recently, Ran and Reurings [20] initiated the study of
hybrid fixed point theorems in partially ordered sets which is further continued in Nieto
and Rodriguez-Lopez [17, 18] and proved the following hybrid fixed point theorems for
the monotone mappings in partially ordered metric spaces using the mixed arguments
from algebra, analysis and geometry.
T HEOREM 1.1. (Nieto and Rodriguez-Lopez [17]) Let (X, ) be a partially ordered set and suppose that there is a metric d in X such that (X, d) is a complete
metric space. Let T : X → X be a monotone nondecreasing mapping such that there
exists a constant k ∈ (0, 1) such that
d(T x, Ty) k d(x, y)
(1)
Mathematics subject classification (2010): 45G10, 45M99, 47H09, 47H10.
Keywords and phrases: Hybrid fixed point theorem, Partially ordered normed linear space, Fractional
integral equation, Existence theorem.
c
, Zagreb
Paper DEA-05-11
155
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BAPURAO C. D HAGE
for all comparable elements x, y ∈ X . Assume that either T is continuous or X is
such that if {xn } is a nondecreasing sequence with xn → x in X , then xn x for all
n ∈ N. Further if there is an element x0 ∈ X satisfying x0 T x0 , then T has a fixed
point which is further unique if “every pair of elements in X has a lower and an upper
bound.”
Another hybrid fixed point theorem in the above direction can be stated as follows.
T HEOREM 1.2. (Nieto and Rodriguez-Lopez [18]) Let (X, ) be a partially ordered set and suppose that there is a metric d in X such that (X, d) is a complete
metric space. Let T : X → X be a monotone nondecreasing mapping satisfying (1). Assume that either T is continuous or X is such that if {xn } is a nonincreasing sequence
with xn → x in X , then xn x for all n ∈ N. Further if there is an element x0 ∈ X
satisfying x0 T x0 , then T has a fixed point which is further unique if “every pair of
elements in X has a lower and an upper bound.”
After the publication of the above two fixed point theorems there is a huge upsurge
in the development of metric fixed point theory in the partially ordered metric spaces.
A good number of fixed and common fixed point theorems have been proved in the
literature for two, three and four mappings in a metric space by modifying the contraction condition (1) suitably as per the requirement of the results. We claim that almost
all the results proved so far along this line though not mentioned have their origin in
a paper due to Heikillä and Lakshmikantham [13]. The main difference is the convergence criteria of sequence of iterations of the monotone mappings under consideration.
The convergence of the sequence in Heikillä and Lakshmikantham [13] is straight forward whereas the convergence of the sequence in Nieto and Rodriguez-Lopez [17] is
due mainly to the metric condition of contraction. The hybrid fixed point theorem of
Heikillä and Lakshmikantham [13] for the monotone mappings in ordered metric spaces
is as follows.
T HEOREM 1.3. (Heikkilä and Lakshmikantham [13]) Let [a, b] be an order interval in a subset Y of the ordered metric space X and let G : [a, b] → [a, b] be a nondecreasing mapping. If the sequence {Gxn } converges in Y whenever {xn } is a monotone
sequence in [a, b], then the well ordered chain of G-iterations of a has the maximum
x∗ which is a fixed point of G. Moreover, x∗ = min{y ∈ [a, b] | Gy y}.
The above hybrid fixed point theorem is applicable in the study of discontinuous
nonlinear equations and has been used throughout the research monograph of Heikkilä
and Lakshmikantham [13]. Note that the convergence of the monotone sequence in
Theorem 1.3 is replaced in Theorems 1.1 and 1.2 by the Cauchy sequence {Gxn }
and completeness of X . Further, the Cauchy nondecreasing sequence is substituted
by equivalent contraction condition for comparable elements in X . Theorems 1.1 and
1.2 are the best hybrid fixed point theorems because they are derived for the mixed arguments from algebra and geometry. The main advantage of Theorems 1.1 and 1.2 is that
the uniqueness of fixed point of the monotone mappings is obtained under certain additional conditions on the domain space such as lattice structure of the partially ordered
H YBRID FIXED POINT THEORY AND APPLICATIONS
157
space under consideration and these fixed point results are then useful in establishing
the uniqueness of the solution of nonlinear differential and integral equations. Again,
some hybrid fixed point theorems of Krasnoselskii type for monotone mappings are
proved in Dhage [5, 6] along the lines of Tarski [21] and Heikillä and Lakshmikantham
[13].
The existence part of Theorems 1.1 and 1.2 may be generalized under weaker
contraction condition as follows.
T HEOREM 1.4. Let (X, ) be a partially ordered set and suppose that there is
a metric d in X such that (X, d) is a complete metric space. Let T : X → X be a
monotone nondecreasing mapping such that there exists a constant k ∈ (0, 1) such that
d(T x, T 2 x) k d(x, T x)
(2)
for all elements x ∈ X comparable to T x ∈ X . Further if T is continuous and there is
an element x0 ∈ X satisfying x0 T x0 or x0 T x0 , then T has a fixed point.
The proof of Theorem 1.4 is essentially same as Theorem 1.1. Note that contraction
condition (2) is weaker than (1) and which is obtained by letting y = T x in the contraction condition (1).
In this paper we again improve the results of Dhage [5, 6] under weaker conditions
and apply them to nonlinear Volterra type fractional integral equations for proving the
existence results under certain monotonic conditions blending with the existence of
either a lower or an upper solution type function for the integral equation under consideration.
2. Basic Hybrid Fixed Point Theorems
Let X be a real vector or linear space. We introduce a partial order in X
as follows. A relation in X is said to be partial order if it satisfies the following
properties:
1. Reflexivity: a a for all a ∈ X ,
2. Antisymmetry: a b and b a implies a = b ,
3. Transitivity: a b and b c implies a c, and
4. Order linearity: x1 y1 and x2 y2 ⇒ x1 + x2 y1 + y2 ;
and x y ⇒ tx ty for t 0 .
The linear space X together with a partial order becomes a partially ordered
linear or vector space. Two elements x and y in a partially ordered linear space X are
called comparable if either the relation x y or y x holds. We introduce a norm
· in a partially ordered linear space X so that X becomes now a partially ordered
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BAPURAO C. D HAGE
normed linear space. If X is complete with respect to the metric d defined through the
above norm, then it is called a partially ordered complete normed linear space.
The following definitions are frequently used in the subsequent part of this paper.
D EFINITION 2.1. A mapping T : X → X is called isotone or nondecreasing if it
preserves the order relation , that is, if x y implies T x Ty for all x, y ∈ X .
D EFINITION 2.2. A mapping ψ : R+ → R+ is called a dominating function or,
in short, D -function if it is an upper semi-continuous and monotonic nondecreasing
function satisfying ψ (0) = 0 .
D EFINITION 2.3. Given a partially ordered normed linear space E , a mapping
Q : E → E is called a partially D -Lipschitz or partially nonlinear D -Lipschitz if
there is a D -function ψ : R+ → R+ satisfying
Qx − Qy ψ (x − y)
(3)
for all comparable elements x, y ∈ E . The function ψ is called a D -function of Q
on E . If ψ (r) = k r , k > 0 , then Q is called partially Lipschitz with the Lipschitz
constant k . In particular, if k < 1 , then Q is called a partially contraction on X with
the contraction constant k . Further, if ψ (r) < r for r > 0 , then Q is called a partially
nonlinear D -contraction with a D -function ψ of Q on X .
The details of different types of contraction definitions appear in the monographs
of Krasnoselskii [14] and Granas and Dugundji [11]. There do exist D -functions and
r
etc. These D the commonly used D -functions are ψ (r) = k r and ψ (r) =
1+r
functions have been widely used in the theory of nonlinear differential and integral
equations for proving the existence and uniqueness results via fixed point methods. See
Browder [3], Deimling [4], Granas and Dugundji [11] and Krasnoselskii [14].
Other notions that we frequently need in what follows are the following definitions.
D EFINITION 2.4. An operator Q on a normed linear space E into itself is called
compact if Q(E) is a relatively compact subset of E . Q is called totally bounded
if for any bounded subset S of E , Q(S) is a relatively compact subset of E . If Q is
continuous and totally bounded, then it is called completely continuous on E .
D EFINITION 2.5. An operator Q on a normed linear space E into itself is called
partially compact if Q(C) is a relatively compact subset of E for all totally ordered
sets or chains C in E . Q is called partially totally bounded if for any totally ordered
and bounded subset C of E , Q(C) is a relatively compact subset of E . If Q is continuous and partially totally bounded, then it is called partially completely continuous
on E .
H YBRID FIXED POINT THEORY AND APPLICATIONS
159
R EMARK 2.1. Note that every compact mapping in a partially normed linear space
is partially compact and every partially compact mapping is partially totally bounded,
however the reverse implications do not hold. Again, every completely continuous
mapping is partially completely continuous and every partially completely continuous
mapping is continuous and partially totally bounded, but the converse may not be true.
Let (X, d) be a metric space and let T : X → X be a mapping. Given an element
x ∈ X , we define an orbit O(x; T ) of T at x by
O(x; T ) = x, T x, T 2 x, ..., T n x, . . . . .
Then T is called T -orbitally continuous on X if for any sequence xn ⊆ O(x; T ), we
have that xn → x∗ implies T xn → T x∗ for each x ∈X. The metric space X is called
T -orbitally complete if every Cauchy sequence xn ⊆ O(x; T ) converses to a point
x∗ in X . Notice that continuity implies that T -orbitally continuity and completeness
implies T -orbitally completeness of a metric space X , but the converse may not be
true.
The following result is frequently used in the analytical fixed point theory of metric
spaces.
L EMMA 2.1. Let ψ : R+ → R+ be a D -function satisfying ψ (r) < r for r > 0 .
Then limn→∞ ψ n (t) = 0 for each t ∈ R+ .
Proof. If t = 0 , then the result follows immediately. So we assume that t > 0 . By
definition of ψ ,
ψ (t) < t =⇒ ψ 2 (t) ψ (t).
By induction principle,
ψ n+1 (t) ψ n (t)
for each n = 0, 1, 2, . . . . Thus {ψ n (t)} is a sequence of positive real numbers which
is bounded below by 0 . Hence, limn→∞ ψ n (t) = d exists. If d = 0 , then by upper
semi-continuity of ψ ,
d = lim ψ n+1 (t) = lim ψ (ψ n (t)) ψ lim ψ n (t) = ψ (d) < d
n→∞
n→∞
n→∞
which is a contradiction.Hence, limn→∞ ψ n (t) = 0 for all t ∈ R+ .
Now we are well equipped to state and prove our main hybrid fixed point results
of this paper. The slight generalizations of Theorems 1.1 and 1.2 are as follows.
T HEOREM 2.1. Let (X, ) be a partially ordered set and let T : X → X be a
nondecreasing mapping. Suppose that there is a metric d in X such that (X, d) is a
T -orbitally complete metric space. Assume that there exists a D -function ψ such that
d(T x, Ty) ψ (d(x, y))
(4)
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BAPURAO C. D HAGE
for all comparable elements x, y ∈ X satisfying ψ (r) < r for r > 0 . Further assume that
either T is T -orbitally continuous on X or X is such that if {xn } is a nondecreasing
sequence with xn → x in X , then xn x for all n ∈ N. If there is an element x0 ∈ X
satisfying x0 T x0 , then T has a fixed point which is further unique if “every pair of
elements in X has a lower and an upper bound.”
Proof. The proof is standard, but for the sake of completeness we give the details
of it. Define a sequence {xn } of successive iterations of T at x0 as
xn+1 = T xn , n = 0, 1, . . . .
(5)
By isotonicity of T , we obtain
x0 x1 · · · xn · · · .
(6)
If xn = xn+1 for some ∈ N, then u = xn is a fixed point of T . Therefore, we
assume that xn = xn+1 for each n ∈ N. If x = xn−1 and y = xn , then by condition (4),
we obtain
(7)
d(xn , xn+1 ) ψ (d(xn−1 , xn ))
for each n = 1, 2, . . . .
Denote dn = d(xn , xn+1 ). Since ψ is a D -function, {dn } is a decreasing sequence
of real numbers which is bounded below by 0 . Hence {dn } is convergent and there
exists a real number d such that
lim dn = d(xn , xn+1 ) = d.
n→∞
We show that d = 0 . If d = 0 , then
d = lim dn = lim d(xn , xn+1 ) lim ψ (d(xn−1 , xn )) ψ (d) < d
n→∞
n→∞
n→∞
which is a contradiction. Hence d = 0 .
We show that {xn } is a Cauchy sequence in X . Suppose not. Then for ε > 0 there
exists a positive integer k n(k) m(k) such that
d(xm(k) , xn(k) ) ε
for all m(k) n(k) k .
Denote rk = d(xm(k) , xn(k) ). Then we have
ε rk = d(xm(k) , xn(k) )
d(xm(k) , xm(k)−1 ) + d(xm(k)−1 , xn(k) )
< dm(k)−1 + ε
H YBRID FIXED POINT THEORY AND APPLICATIONS
161
and so, limk→∞ rk = ε .
Again,
ε rk = d(xm(k) , xn(k) )
d(xm(k) , xm(k)+1 ) + d(xm(k)+1 , xn(k)+1 ) + d(xn(k)+ , xn(k) )
dm(k) + ψ (rk ) + dn(k).
Taking the limit as k → ∞, we obtain
ε ψ (ε ) < ε .
which is a contradiction. Therefore {xn } is a Cauchy sequence in X . The metric space
(X, d) being T -orbitally complete, there is a point x∗ ∈ X such that
lim T n x0 = lim xn = x∗ .
n→∞
n→∞
Suppose that T is T -orbitally continuous on X . Then,
T x∗ = T lim T n x0 = lim T n+1 x0 = x∗ .
n→∞
Next, suppose that
of ψ , we obtain
T nx
0
= xn n→∞
x∗
for all n ∈ N. Then by upper semi-continuity
d(T x∗ , x∗ ) d(T x∗ , T n+1 x0 ) + d(T n+1 x0 , x∗ )
ψ (d(x∗ , T n x0 )) + ψ (d(T n x0 , x∗ ))
ψ ( lim d(x∗ , T n x0 )) + ψ lim d(T n x0 , x∗ )
n→∞
n→∞
= 0.
Hence, T x∗ = x∗ . Thus, in both the cases T has a fixed point.
To prove uniqueness, let y∗ be another fixed point of T in X and suppose that
every pair of elements in X has a lower and an upper bound. Then there is a point
z ∈ X such that either z x and z y, or x z and y z. Thus in both the cases, by
Lemma 2.1, we have
d(x∗ , y∗ ) d(x∗ , T n z) + d(T n z, y∗ )
= d(T n x∗ , T n z) + d(T n z, T n y∗ )
ψ n d(x∗ , z) + ψ n d(z, y∗ )
= 0 as n → ∞.
This shows that T has a unique fixed point and the proof of the theorem is complete.
As a special case of our Theorem 2.1 we obtain Theorem 1.1 of Nieto and RodriguezLopez [17] for partially contraction mappings in partially ordered complete metric
spaces.
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BAPURAO C. D HAGE
Sometimes it possibles that a mapping T is not a nonlinear D -contraction, but
some iterations of it is a nonlinear D -contraction on X . In this connection the following hybrid fixed point theorem is noteworthy.
T HEOREM 2.2. Let (X, ) be a partially ordered set and let T : X → X be a
nondecreasing mapping. Suppose that there is a metric d in X such that (X, d) is a
T -orbitally complete metric space. Assume that there exists a D -function ψ and a
positive integer p such that
d(T p x, T p y) ψ (d(x, y))
(8)
for all comparable elements x, y ∈ X satisfying ψ (r) < r for r > 0 . Further assume that
either T is T -orbitally continuous on X or X is such that if {xn } is a nondecreasing
sequence with xn → x in X , then xn x for all n ∈ N. If there is an element x0 ∈ X
satisfying x0 T x0 and “every pair of elements in X has a lower and an upper bound,”
then T has a unique fixed point.
Proof. Set S = T p . Then S : X → X is a S -orbitally continuous nondecreasing
mapping. Also there exists the element x0 ∈ X such that x0 Sx0 . Now, an application
of Theorem 2.1 yields that S has a unique fixed point, that is, it is a point u ∈ X such
that S(u) = T p (u) = u . Now T (u) = T (T p u) = S(Tu), showing that Tu is again a
fixed point of S . By the uniqueness of u, we get Tu = u. The proof is complete. T HEOREM 2.3. Let (X, ) be a partially ordered set and let T be a nondecreasing mapping. Suppose that there is a metric d in X such that (X, d) is a T -orbitally
complete metric space. Assume that there exists a D -function ψ satisfying he contractive condition (4). Further assume that either T is T -orbitally continuous on X or X
is such that if {xn } is a nonincreasing sequence with xn → x in X , then xn x for all
n ∈ N. If there is an element x0 ∈ X satisfying x0 T x0 , then T has a fixed point which
is further unique if “every pair of elements in X has a lower and an upper bound.”
Proof. The proof is similar to Theorem 2.1 and therefore we omit the details.
T HEOREM 2.4. Let (X, ) be a partially ordered set and let T : X → X be nondecreasing mapping. Suppose that there is a metric d in X such that (X, d) is a
T -orbitally complete metric space. Assume that there exists a D -function ψ and a
positive integer p satisfying he contractive condition (8). Assume further that either T
is T -orbitally continuous on X or X is such that if {xn } is a nondecreasing sequence
with xn → x in X , then xn x for all n ∈ N. If there is an element x0 ∈ X satisfying
x0 T x0 and “every pair of elements in X has a lower and an upper bound,” then T
has a unique fixed point.
R EMARK 2.2. The hypothesis concerning the conditions x0 T x0 and x0 T x0
in Theorems 2.2 and 2.4 may be replaced by the hypothesis with weaker conditions
x0 T p x0 and x0 T p x0 respectively.
H YBRID FIXED POINT THEORY AND APPLICATIONS
163
R EMARK 2.3. The monotone hypothesis of the mapping T together with the condition x0 T x0 or x0 T p x0 for some x0 ∈ X in all above hybrid fixed point theorems
may be replaced with the dominating character of T , that is, either x T x or x T x
for all x ∈ X .
R EMARK 2.4. The convergence condition that X is such that if {xn } is a nondecreasing sequence with xn → x in X , then xn x for all n ∈ N, holds in particular if
X is a normed linear space and the order relation is defined in X through the order
cones K in it which is defined later in Section 3. The hybrid fixed point theorems with
this convergence condition is useful in the study of nonlinear differential and integral
equations with discontinuous nonlinearities.
R EMARK 2.5. The conclusion of Theorems 2.1 and 2.3 remains true if we replace
the contractive condition (4) with the following condition of generalized contraction ,
1
d(T x, Ty) ψ max d(x, y), d(x, T x), d(y, Ty), d(x, Ty) + d(y, T x)
(9)
2
and the conclusion of Theorems 2.2 and 2.4 also remains true if we replace the contractive condition (8) with the following generalized contraction condition,
1
p
p
p
p
p
p
d(T x, T y) ψ max d(x, y), d(x, T x), d(y, T y), d(x, T y) + d(y, T x)
.
2
(10)
The contraction conditions of the type (9) have been employed in several fixed point
theorems for the mappings in partially ordered metric spaces, however to the best of
our knowledge, the contraction condition of the type (10) has never been used in the
fixed point theorems in ordered spaces.
R EMARK 2.6. The argument that every pair of elements in an ordered set X has
a lower and an upper bound holds if (X, ) is a lattice. The details about the ordered
sets and the lattice structure may be found in Birkhoff [2].
Note that Theorems 2.1, 2.2, 2.3 and 2.4 have some nice applications to various
nonlinear problems modeled on nonlinear equations for proving the existence as well as
uniqueness of the solutions under generalized Lipschitz conditions. The following two
fixed point theorems are perhaps new to the literature. The basic principle in formulating these theorems is the same as that of Dhage [5] and Nieto and Rodriguez-Lopez
[17]. Before stating these results we give a useful definition.
D EFINITION 2.6. The order relation and the metric d in a non-empty set X
are said to be compatible if {xn } is a monotone (order preserving), that is, monotone
nondecreasing sequence in X and if a subsequence {xnk } of {xn } converges to x∗ implies that the whole sequence {xn } converges to x∗ . Similarly, given a partially ordered
normed linear space (X, , · ), the order relation and the norm · are said to be
compatible if and the metric d defined through the norm · are compatible.
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BAPURAO C. D HAGE
Clearly, the set R of real numbers with usual order relation and the norm defined by the absolute value function has this property. Similarly, the space C(J, R) with
usual order relation defined by x y if and only if x(t) y(t) for all t ∈ J and the
usual standard supremum norm · defined by x = sup |x(t)| are compatible.
t∈J
T HEOREM 2.5. Let X be a partially ordered linear space and suppose that there
is a norm in X such that X is a normed linear space. Let T : X → X be a nondecreasing, partially compact and continuous mapping. Further if the order relation and the norm · in X are compatible and if there is an element x0 ∈ X satisfying
x0 T x0 , then T has a fixed point.
Proof. Define a sequence {xn } of successive iterations of T at x0 as
xn+1 = T xn , n = 0, 1, 2, . . . .
(∗)
Since T is nondecreasing and x0 T x0 , one has
x0 x1 · · · xn · · · .
(∗∗)
Denote C = {x0 , x1 . . . , xn , . . .} . Then C is totally ordered set in the normed linear
space X and from the partially compactness of T , it follows that T (C) is relatively
compact subset of X . As a result {x1 , . . . , xn , . . .} ⊆ T (C) and the sequence {xn } has
a convergent subsequence {xnk } converging to some point x∗ in X . From the compatibility of the order relation and the norm · , it follows that the whole sequence
sequence {xn } converges to x∗ . Finally, by the continuity of T , we obtain
T x∗ = T lim xn = lim T xn = lim xn+1 = x∗ .
n→∞
n→∞
n→∞
This completes the proof.
T HEOREM 2.6. Let X be a partially ordered linear space and suppose that there
is a norm in X such that X is a normed linear space. Let T : X → X be a nondecreasing, partially compact and continuous mapping. Further if the order relation and the norm · in X are compatible and if there is an element x0 ∈ X satisfying
x0 T x0 , then T has a fixed point.
Proof. The proof is similar to Theorem 2.4 and we omit the details.
R EMARK 2.7. The hypothesis of continuity on the mapping T in above Theorems
2.4 and 2.5 may be replaced with a weaker T -orbitally continuity condition of the
mapping T on X .
In the following section, we try to combine Theorem 2.1 and Theorem 2.5 to yield
some Krasnoselskii type hybrid fixed point theorems in a partially ordered complete
normed linear space and discuss some of their applications to nonlinear Volterra type
fractional integral equations.
H YBRID FIXED POINT THEORY AND APPLICATIONS
165
3. Krasnoselskii Type Fixed Point Theorems
Here, in this section, first we list different Krasoselskii [14] type fixed point theorems according to their degree of generality available in the literature. The following
version of Krasnoselskii’s fixed point theorem due to Dhage [8] is known in the literature which do not need any order structure of the Banach space under consideration
and has some nice applications to nonlinear linearly perturbed differential and integral
equations.
T HEOREM 3.1. (Dhage [8]) Let S be a closed convex and bounded subset of a
Banach space X and let A : X → X and B : S → X be two operators satisfying the
following conditions:
(a) A is nonlinear D -contraction,
(b) B is completely continuous, and
(c) Ax + By = x for all y ∈ S =⇒ x ∈ S .
Then the operator equation
Ax + Bx = x
(11)
has a solution.
Theorem 3.1 is very much useful and has been applied to linear perturbations
of differential and integral equations by several authors in the literature for proving
the existence of the solutions under mixed Lipschitz and compactness type conditions.
Here we do not need any other structure of the Banach space under consideration. See
Dhage [7], Dhage and Jadhav [9] and the references cited therein.
The theory of Krasnoselskii type fixed point theorems using the order relation is
initiated by the present author in Dhage [6] and developed further in a series of papers.
See Dhage [5, 6, 7, 8] and the references therein. Below we state these hybrid fixed
point theorems as per their degree of generality. The following Krasnoselskii type fixed
point theorem in a complete lattice is proved in Dhage [5].
T HEOREM 3.2. (Dhage [5]) Let S be a non-empty closed subset of a complete
lattice Banach space X and let A : S → X and A : S → X be two operators satisfying
the following conditions:
(a) A is nonlinear D -contraction,
(b) B is compact and continuous,
(c) Ax + By ∈ S for all x, y ∈ S , and
(c) (I − A)−1 B is isotonic increasing on S , where I denotes the identity mapping on
X.
Then the operator equation (11) has a least and a greatest solution in S.
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BAPURAO C. D HAGE
Other hybrid fixed point theorems concerning the solution of operator equation
(11) which make use of order structure of Banach space under consideration are as
follows. Before stating these hybrid fixed point theorems, we give a useful definition.
D EFINITION 3.1. A non-empty closed subset K of a normed linear space X is
called an order cone if the following properties are satisfied.
(i) K + K ⊆ K ,
(ii) λ K ⊆ K , and
(iii) {−K } ∩ K = θ , where θ is the zero element of X .
The ordered Banach space X together with the order cone K is denoted by
(X, K ). The details of order cones and their properties appear in Heikkilä and Lakshmikantham [13] and the references given therein. We define the order relation in X
as follows. For any x, y ∈ X , we define
x y ⇐⇒ y − x ∈ K .
(∗)
The following lemma is sometimes useful in the study of discontinuous fractional differential and integral equations.
L EMMA 3.1. If {xn } is a nondecreasing sequence in the ordered Banach space
(X, K ) converging to the point x ∈ X , then xn x for all n ∈ N.
Proof. Since {xn } is nondecreasing, we have
x1 x2 · · · xn · · · ;
or
xm xn
∀ m n ∈ N.
By definition of , we get
xm − xn ∈ K ∀ m n ∈ N.
As the order cone K is closed, one has
lim xm − xn = lim xm − xn = x − xn ∈ K
m→∞
m→∞
which implies that xn x for each n ∈ N. The proof of the lemma is complete.
Let x0 , y0 ∈ X be fixed elements with x0 y0 . Then by an order interval or
vector segment we mean a set [x0 , y0 ] in X defined by
[x0 , y0 ] = {x ∈ X | x0 x y0 }.
Clearly, [x0 , y0 ] is a closed, convex set in X which is further bounded if the order
cone K in X is normal. A few details of the order cones and their properties, a reader
is referred to Heikkilä and Lakshmikatham [13] and the references therein.
H YBRID FIXED POINT THEORY AND APPLICATIONS
167
T HEOREM 3.3. (Dhage [6]) Let [x0 , y0 ] be the order interval in an ordered Banach space X and let A : X → X and B : [x0 , y0 ] → X be two nondecreasing operators
satisfying the following conditions:
(a) A is nonlinear D -contraction,
(b) B is compact and continuous, and
(c) Ax + By ∈ [x0 , y0 ] for all x, y ∈ [x0 , y0 ].
If the order cone K in a Banach space X is normal, then the operator equation (11)
has a least and a greatest solution in [x0 , y0 ].
Our last known Krasnoselskii type hybrid fixed point theorem which is weaker
version than previous two versions is the following result.
T HEOREM 3.4. (Dhage [6]) Let [x0 , y0 ] be an order interval in an ordered Banach space X and let A, B : [x0 , y0 ] → X be two operators satisfying the following
conditions:
(a) A is nonlinear D -contraction,
(b) B is compact and continuous,
(c) Ax + Bx ∈ [x0 , y0 ] for all x ∈ [x0 , y0 ], and
(d) (I − A)−1 B is monotone nondecreasing on [x0 , y0 ].
If the order cone K in a Banach space X is normal, then the operator equation (11)
has a least and a greatest solution in [x0 , y0 ].
R EMARK 3.1. Note that (I − A)−1 B is nondecreasing if A and B are nondecreasing but the converse may not be true. Therefore, Theorem 3.4 is more general hybrid
fixed point theorem than any other so far discussed Krasnoselskii type fixed point theorem in an ordered Banach space X in view of hypothesis (c).
Here, we obtain other interesting versions of Krasnoselskii type fixed point theorem in a partially ordered complete normed linear space under weaker hypotheses (a),
(b) and (c) of Theorems 3.1 through 3.4.
T HEOREM 3.5. Let X, , · be a partially ordered complete normed linear
space such that the order relation and the norm · in X are compatible. Let
A, B : X → X be two nondecreasing operators such that
(a) A is continuous and partially nonlinear D -contraction,
(b) B is continuous and partially compact,
(c) there exists an element x0 ∈ X such that x0 Ax0 + By for all y ∈ X , and
168
BAPURAO C. D HAGE
(d) every pair of elements x, y ∈ X has a lower and an upper bound in X .
Then the operator equation (11) has a solution in X .
Proof. Define the operator T : X → X by
T = (I − A)−1 B.
(12)
Clearly, the operator T is well defined. To see this, let y ∈ X be fixed and define
a mapping Ay : X → X by
Ay (x) = Ax + By.
(13)
Ay is nondecreasing and by hypothesis (c), there is a point x0 ∈ X such that x0 T x0 .
Now, for any two comparable elements x1 , x2 ∈ X , one has
Ay (x1 ) − Ay(x2 ) = Ax1 − Ax2 ψ (x1 − x2 )
(14)
where, ψ is a D -function of T on X . So Ay is a partially nonlinear D -contraction on
X . Hence, by an application of a fixed point Theorem 2.1, Ay has a unique fixed point,
say x∗ ∈ X . Thus, we have a unique element x∗ in X such that
Ay (x∗ ) = Ax∗ + By = x∗
which implies that
or,
(I − A)−1 By = x∗
Ty = x∗ .
Thus the mapping T : X → X is well defined. Now define a sequence {xn } of iterates
of T at x0 , that is, xn+1 = T xn for n = 0, 1, 2, .... From hypothesis (c), it follows
that x0 T x0 . Again, by Remark 3.1, we obtain that the mapping T is monotone
nondecreasing on X . So we have that
x0 x1 x2 · · · xn · · · .
(15)
Since B is partially compact and (I − A)−1 is continuous, the composition mapping T = (I − A)−1 B is partially compact and continuous on X into X . Therefore the
sequence {xn } has a convergent subsequence converging to some point, say x∗ ∈ X and
from the compatibility of order relation and the norm · in X , it follows that the
whole sequence {xn } converges to the point x∗ in X. Hence, an application of Theorem 2.5 implies that T has a fixed point. This further implies that (I − A)−1 Bx∗ = x∗
or Ax∗ + Bx∗ = x∗ . This completes the proof.
T HEOREM 3.6. Let X, , · be a partially ordered complete normed linear
space such that the order relation and the norm · in X are compatible. Let
A, B : X → X be two nondecreasing mappings satisfying
H YBRID FIXED POINT THEORY AND APPLICATIONS
169
(a) A is linear and bounded and A p is partially nonlinear D -contraction for some
positive integer p ,
(b) B is continuous and partially compact,
(c) there exists an element x0 ∈ X such that x0 Ax0 + By for all y ∈ X .
(d) every pair of elements x, y ∈ X has a lower and an upper bound in X .
Then the operator equation (11) has a solution in X .
Proof. Define the operator T on X by
T = (I − A)−1 B.
(16)
Now the mapping (I − A)−1 exists in view of the relation
(I − A)−1 = (I − A p )−1
p−1
∑ A j,
(17)
j=1
p−1
where
∑ Aj
j=1
is bounded and (I − A p )−1 exists in view of a Theorem 2.2 . Hence,
(I − A)−1 exists and is continuous on X .
Next, the operator T is well defined. To see this, let y ∈ X be fixed and define a
mapping Ay : X → X by
Ay (x) = Ax + By.
(18)
Clearly, Ay is nondecreasing on X into itself. Again, by hypothesis (c), there is a point
x0 ∈ X such that
x0 Ay (x0 ) A2y (x0 ) . . . Ayp (x0 ).
Now for any two comparable elements x1 , x2 ∈ X , one has
Ayp (x1 ) − Ayp(x2 ) ψ (x1 − x2 ).
Hence, by Theorem 2.2, there exists a unique element x∗ such that
Ayp (x∗ ) = A p (x∗ ) + By = x∗ .
This further implies that Ay (x∗ ) = x∗ and x∗ is a unique fixed point of Ay . Thus, we
have Ay (x∗ ) = x∗ = Ax∗ + By, or, (I − A)−1 By = x∗ . As a result, Ty = x∗ and so T is
well defined. The rest of the proof is similar to Theorem 3.5 and we omit the details.
The proof is complete.
Below we state two hybrid fixed point theorems in a partially ordered complete
normed linear space by reverting the inequality given in hypothesis (c) of the above
two fixed point theorems. The proofs of these theorems are similar to Theorems 3.5
and 3.6. Hence we omit the details.
170
BAPURAO C. D HAGE
T HEOREM 3.7. Suppose that X, , · is a partially ordered complete normed
linear space such that the order relation and the norm · in X are compatible. Let
A, B : X → X be two nondecreasing operators such that
(a) A is continuous and partially nonlinear D -contraction,
(b) B is continuous and partially compact,
(c) there exists an element x0 ∈ X such that Ax0 + By x0 for all y ∈ X , and
(d) every pair of elements x, y ∈ X has a lower and an upper bound in X .
Then the operator equation (11) has a solution.
T HEOREM 3.8. Let X, , · be a partially ordered complete normed linear
space such that the order relation and the norm · in X are compatible. Let
A, B : X → X be two nondecreasing mappings such that
(a) A is linear and bounded and A p is partially nonlinear D -contraction for some
positive integer p ,
(b) B is a continuous and partially compact,
(c) there exists an element x0 ∈ X such that Ax0 + By x0 for all y ∈ X ,
(d) every pair of elements x, y ∈ X has a lower and an upper bound in X .
Then the operator equation (11) has a solution.
R EMARK 3.2. The hypothesis (d) of Theorems 3.5 and 3.6 holds if the partially
ordered set X is a lattice. Furthermore, the space C(J, R) of continuous real-valued
functions on the closed and bounded interval J = [a, b] is a lattice, where the order
relation is defined as follows. For any x, y ∈ C(J, R), x y if and only if x(t) y(t)
for all t ∈ J . The real variable operations show that min{x, y} and max{x, y} are
respectively the lower and upper bounds for the pair of elements x and y in X .
4. Fractional Integral Equations
In this section we apply the hybrid fixed point theorems proved in the previous
two sections to some nonlinear fractional integral equations Volterra type for proving
the existence and uniqueness theorems under certain mixed arguments from algebra,
geometry and topology. First we deal with the nonlinear Volterra fractional integral
equations for existence as well as uniqueness results under certain mixed conditions.
Let us recall some basic definitions of fractional calculus [15, 19] which we need in
what follows.
H YBRID FIXED POINT THEORY AND APPLICATIONS
171
D EFINITION 4.1. The Riemann-Liouville fractional derivative of a n -times differential function x : J → R of fractional order q is defined as
n t
1
d
Dq0+ x(t) =
(t − s)n−q−1x(s) ds, n − 1 < q < n, n = [q] + 1,
Γ(n − q) dt
0
provided that the integral exists, where [q] denotes the integer part of the real number
q.
Similarly, Caputo fractional derivative of a n -times differential function x : J →
R fractional order q is defined as
c
q
D0+ x(t) =
1
Γ(n − q)
t
0
(t − s)n−q−1x(n) (s) ds, n − 1 < q < n, n = [q] + 1,
provided that the integral exists, where [q] denotes the integer part of the real number
q.
D EFINITION 4.2. The Riemann-Liouville fractional integral of a function x :
J → R of order q > 0 is defined as
q
x(t) =
I0+
1
Γ(q)
t
0
x(s)
ds,
(t − s)1−q
provided the integral exists.
The equation involving the unknown function and its fractional derivatives is called
a fractional differential equation and an equation involving the presence of unknown
function under fractional integral is called the fractional integral equation. The topic of
fractional differential and integral equation is of current interest and several authors all
over the contributed to this area of investigation. The details of fractional calculus and
fractional differential equations may be found in Kilbas et. al. [15], Lakshmikantham
et al.[16] and Podlubny [19] etc. In the following section we discuss some Volterra type
fractional integral equations for the existence results via the abstract results developed
in the previous sections.
4.1. The Volterra fractional integral equations
Given 0 < q < 1 and p = 1 − q , and given a closed and bounded interval J =
[t0 ,t0 + a] in R, denote, by C(J, R) and C p (J, R) the spaces of continuous real-valued
functions on J and
(19)
C p (J, R) = u ∈ C(J, R) | (t − t0 ) p u(t) ∈ C(J, R) .
Consider the following nonlinear Volterra fractional integral equation (in short
VFIE)
t
1
(t − s)q−1 g(s, x(s)) ds
(20)
x(t) = h(t) +
Γ(q) t0
172
BAPURAO C. D HAGE
for all t ∈ J , where Γ is a gamma function, h : J → R and g : J × R → R is continuous
functions.
x0 (t − t0 )q−1
, where x0 = x(t)(t − t0 )1−q , we
The special case when h(t) =
Γ(q)
t=t0
obtain the Volterra fractional integral equation considered in Lakshmikantham et al.[16].
Thus, equation (19) is a more general Volterra type fractional integral equation that has
been studied in the literature on fractional calculus and applications.
By a solution of the VFIE (19) we mean a function x ∈ C p (J, R) that satisfies the
equation (19) on J .
In the following we shall discuss the basic existence result for the VFIE (19) on
J . We need the following set of assumptions in what follows.
(H 1 ) g is bounded on J × R with bound Mg .
(H 2 ) g(t, x) is nondecreasing in x for each t ∈ J .
(H 3 ) There exists an element u0 ∈ C(J, R) such that
u0 (t) h(t) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, u0 (s)) ds
for all t ∈ J .
Note that all the hypotheses (H 1 ) through (H 3 ) are standard and frequently used
in the theory of nonlinear differential and integral equations.
T HEOREM 4.1. Assume that the hypotheses (H 1 ) through (H 3 ) hold. Then the
VFIE (19) has a solution defined on J .
Proof. Set X = C(J, R) with the supremum norm · defined by
x = sup |x(t)|.
t∈J
(∗∗)
Define an order relation in X as follows. Let any x, y ∈ X . Then x y if and
only if x(t) y(t) for all t ∈ J . Clearly, as mentioned earlier, C(J, R) is a lattice.
Therefore, the lower and upper bounds for every pair of elements x, y ∈ C(J, R) exist.
Furthermore, the order relation and the norm · in X are compatible. Define the
operator T on X by
T x(t) = h(t) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, x(s)) ds, t ∈ J.
(21)
From the continuity of the functions involved in (21) it follows that T defines
a mapping T : X → X . We show that the operator T satisfies all the conditions of
Theorem 2.5 on the Banach space X . First we note that x0 T x0 in view of hypothesis
H YBRID FIXED POINT THEORY AND APPLICATIONS
173
(H 3 ). Therefore, it is enough to that T ia a compact and continuous operator on X in
view of Remark 2.4. This will be done in the following three steps:
Step 1: T is uniformly bounded .
First we show that T is a uniformly bounded on C(J, R). Let x ∈ C(J, R) be an
arbitrary element. Then
t
1
(t − s)q−1 |g(s, x(s))| ds
Γ(q) t0
Mg t
|h(t)| +
(t − s)q−1 ds
Γ(q) t0
Mg
h +
(t − t0 )q
qΓ(q)
Mg aq
=M
h +
Γ(q + 1)
|T x(t)| |h(t)| +
for all t ∈ J and for all x ∈ X . This shows that T is a uniformly bounded operator on
X.
Step 2: T (X) is an equicontinuous set.
Let x ∈ X be arbitrary. Then, by definition of T , we have
|T x(t1 ) − T x(t2 )| h(t1 ) − h(t2)
t2
1 t1
1
q−1
q−1
+
(t1 − s) g(s, x(s)) ds −
(t2 − s) g(s, x(s)) ds
Γ(q) t0
Γ(q) t0
h(t1 ) − h(t2)
t2
1 t1
q−1
q−1
(t1 − s) g(s, x(s)) ds − (t1 − s) g(s, x(s)) ds
+
Γ(q) t0
t0
t2
1 t2
q−1
q−1
+
(t1 − s) g(s, x(s)) ds −
(t2 − s) g(s, x(s)) ds
Γ(q) t0
t0
h(t1 ) − h(t2)
1 t2
q−1
(t1 − s) |g(s, x(s))| ds
+
Γ(q) t1
t0 +a 1
(t1 − s)q−1 − (t2 − s)q−1 |g(s, x(s))| ds
+
Γ(q) t0
h(t1 ) − h(t2)
1 t2
q−1
(t1 − s) Mg ds
+
Γ(q) t1
t0 +a 1
(t1 − s)q−1 − (t2 − s)q−1 Mg ds +
Γ(q) t0
Mg
|t1 − t2 |q
h(t1 ) − h(t2) +
Γ(q + 1)
174
BAPURAO C. D HAGE
+
Mg
Γ(q)
t0 +a t1
(t1 − s)q−1 − (t2 − s)q−1 = h(t1 ) − h(t2) +
Mg
|t1 − t2 |q
Γ(q + 1)
Mg t0 +a +
(t1 − s)q−1 − (t2 − s)q−1 ds
Γ(q) t0
→ 0 as t1 → t2
uniformly for all t1 ,t2 ∈ J . This shows that the family T (X) is an equicontinuous set
in X . Now we apply Arzelá-Ascoli theorem to yield that T is a compact operator on
X.
Step 3 : T is continuous.
Let {xn } be a sequence of points in X converging to a point x in X . Then, by
dominated convergence theorem and the continuity of the function g , we obtain
n→∞
t
1
lim (t − s)q−1 g(s, xn (s)) ds
Γ(q) n→∞ t0
t
1
= h(t) +
(t − s)q−1 lim g(s, xn (s)) ds
n→∞
Γ(q) t0
t
1
(t − s)q−1 g(s, x(s)) ds
= h(t) +
Γ(q) t0
lim T xn (t) = h(t) +
= T x(t)
for all t ∈ J . This shows that T x is a pointwise continuous function on J . Since T (X)
is equicontinuous, the sequence {T xn } is equicontinuous. Now, it can be shown as in
Guenther et.al. [12] (see aslo Granas and Dugundji [11]) that {T xn } converges to T x
uniformly on J . Hence T a is continuous operator on X . Thus all the conditions of
Theorem 2.5 are satisfied by the operator T on X . Hence we apply Theorem 2.5 and
conclude that T has a fixed point in X . This further implies that the VFIE (19) has a
solution defined on J and the proof of the theorem is complete.
Our next result is about the uniqueness theorem for the VFIE (19) under a certain
partial Lipschitz condition called the one-sided Lipschitz condition together with the
application of a basic hybrid fixed point theorem. We need the following hypothesis in
the sequel.
(B 1 ) There exist constants L > 0 and K > 0 such that
0 g(t, x) − g(t, y) L(x − y)
K + (x − y)
for all x, y ∈ R with x y.
T HEOREM 4.2. Assume that the hypotheses (B 1 ) and (H 2 )-(H 3 ) hold. Further,
La
K , then the VFIE (19) has a unique solution defined on J .
if
Γ(q + 1)
175
H YBRID FIXED POINT THEORY AND APPLICATIONS
Proof. Set X = C(J, R) with supremum norm · defined by x = sup |x(t)|.
t∈J
Define an order relation in X as follows. Let any x, y ∈ X . Then x y if and
only if x(t) y(t) for all t ∈ J . Clearly, as mentioned earlier, C(J, R) is a lattice.
Therefore, the lower and upper bounds for every pair of elements x, y ∈ C(J, R) exist.
Define an operator T on X into itself by (21). From hypothesis (H 2 ) it follows that
T is a monotone nondecreasing operator on X . We show that T is a partial nonlinear
D -contraction on X . Let x, y ∈ X be such that x y. Then, by hypothesis (B 1 ), we
obtain
t
t
T x(t) − Ty(t) = 1 (t − s)q−1 g(s, x(s)) ds − (t − s)q−1 g(s, y(s)) ds
Γ(q) t0
t0
t
1
(t − s)q−1g(s, x(s)) − g(s, y(s)) ds
Γ(q) t0
t
L||x − y||
1
ds
(t − s)q−1
Γ(q) t0
K + ||x − y||
La
||x − y||
·
(22)
Γ(q + 1) K + ||x − y||
Taking supremum over t , we obtain
T x − Ty La
||x − y||
·
= ψ (||x − y||)
Γ(q + 1) K + ||x − y||
for all x, y ∈ X with x y, where ψ (r) =
(23)
r
La
·
< r for r > 0 since
Γ(q + 1) K + r
La
K . Clearly, ψ is a D -function for the operator A on X . Consequently, A
Γ(q + 1)
is a partially D -nonlinear contraction on X. Now the desired conclusion follows by a
direct application of Theorem 2.1.
If the nonlinearity g involved in VFIE (19) is not continuous on the domain of its
definition, then we consider the following hypothesis:
(B 2 ) The function s → (t − s)q−1 g(s, x(s)) is Riemann integrable on J for each x ∈
C(J, R) with t > s.
T HEOREM 4.3. Assume that the hypotheses (B 1 ) and (H 3 ) hold. Suppose that the
La
K
function g is not continuous but the hypothesis (B 2 ) holds. Further if
Γ(q + 1)
holds, then the VFIE (19) has a unique solution defined on J .
Proof. Set X = C(J, R) and define an order relation and the norm · in X as
in the proof of Theorem 4.2. Define the order cone KC in the Banach space X as
KC = x ∈ X | x(t) 0 for all t ∈ J .
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BAPURAO C. D HAGE
Then the order relation is identical with the order relation defined by (**) with the
help of order cone KC in X . The rest of the proof is similar to that of Theorem 4.2
and now the conclusion follows directly by an application of Lemma 3.1.
Notice that the Reimann integrability of the function s → (t − s)q−1 g(s, x(s)) for
t > s is guaranteed in the following three cases.
1. g is continuous on J × R.
2. g(s, x) is Carathéodory, i.e., g(s, x) is measurable in s for each x ∈ R and continuous in x for each t ∈ J and it is bounded on J × R.
3. The map s → g(s, x) is monotone increasing for all x ∈ R.
We mention that above stated cases have been discussed in the literature on the
theory of nonlinear continuous and discontinuous differential equations.
R EMARK 4.1. The conclusion of Theorems 4.1, 4.2 and 4.4 holds if we replace
the hypothesis (H 3 ) by the following one.
(H 3 ) There exists an element u0 ∈ C(J, R) such that
u0 (t) h(t) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, u0 (s)) ds.
for all t ∈ J .
4.2. Fractional integral equations of mixed type
Given a closed and bounded interval J = [t0 ,t0 + a] in R, R the set of real numbers, for some t0 ∈ R and a ∈ R with a > 0 and given a real number 0 < q < 1 ,
consider the nonlinear hybrid fractional integral equation (in short HFIE)
1
x(t) = f (t, x(t)) +
Γ(q)
t
t0
(t − s)q−1 g(s, x(s)) ds, t ∈ J,
(24)
where f : J × R → R is continuous and g : J × R → R is locally Hölder continuous.
We seek the solutions of HFIE (24) in the space C(J, R) of continuous real-valued
functions defined on J . We consider the following set of hypotheses in what follows.
(H 4 ) There exist constants L > 0 and K > 0 such that
0 f (t, x) − f (t, y) for all x, y ∈ R with x y. Moreover L K.
L(x − y)
K + (x − y)
H YBRID FIXED POINT THEORY AND APPLICATIONS
177
(H 5 ) There exists an element u0 ∈ X = C(J, R) such that
u0 (t) f (t, u0 (t)) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, y(s))ds
for all t ∈ J and y ∈ X .
R EMARK 4.2. The condition given in hypothesis (H 5 ) is a little more than that of
a lower solution for the HFIE (24). It is clear that u0 is a lower solution of the FIE (24),
however the converse is not true.
T HEOREM 4.4. Assume that hypotheses (H 1 ) through (H 5 ) hold. Then the HFDE
(24) admits a solution.
Proof. Set X = C(J, R), the Banach space of continuous real-valued functions on
J with usual supremum norm · given by x = sup |x(t)|.
t∈J
Define an order relation in X as follows. Let any x, y ∈ X . Then x y if and
only if x(t) y(t) for all t ∈ J . Clearly, as mentioned earlier, X is a lattice. Therefore,
the lower and upper bounds for every pair of elements x, y ∈ X exist. Furthermore,
the order relation and the norm · in X are compatible. Define two operators
A, B : X → X by
Ax(t) = f (t, x(t)), t ∈ J
(25)
and
Bx(t) =
1
Γ(q)
t
t0
(t − s)q−1 g(s, x(s))ds, t ∈ J.
(26)
Then the given hybrid fractional integral equation (24) is transformed into an
equivalent operator equation as
Ax(t) + Bx(t) = x(t), t ∈ J.
(27)
From the continuity of the functions involved in the right hand side of (25) and
(26) it follows that A and B define the operators A, B : X → X . We show that the
operators A and B satisfy all the conditions of Theorem 3.5 on X. First, we show that
A is a nonlinear contraction on X. Let x, y ∈ X be such that x y. Then, by hypothesis
(H 4 ), we obtain
|Ax(t) − Ay(t)| = | f (t, x(t)) − f (t, y(t))|
L|x(t) − y(t)|
K + |x(t) − y(t)|
L||x − y||
.
K + ||x − y||
(28)
for all t ∈ J . Taking the supremum over t , we obtain
Ax − Ay L||x − y||
= ψ (||x − y||)
K + ||x − y||
(29)
178
BAPURAO C. D HAGE
Lr
< r for r > 0. Clearly, ψ is a
K +r
D -function for the operator A on X. Consequently, A is a partially nonlinear D contraction on X.
for all x, y ∈ X with x y, where ψ (r) =
Next, we show that B is a compact continuous operator on X. To finish, we show
that B(X) is a uniformly bounded and equi-continuous set in X. Now for any x ∈ X ,
one has
t
1
|t − s|q−1 |g(s, x(s))| ds
Γ(q) t0
Mg t
(t − s)q−1 ds
Γ(q) t0
aq Mg
Γ(q + 1)
|Bx(t)| (30)
for all t ∈ J which shows that B is a uniformly bounded set in X . Now, let t1 ,t2 ∈ J be
arbitrary. Then,
|Bx(t1 ) − Bx(t2 )| Mg
Γ(q)
+
→0
t0
|(t1 − s)q−1 − (t2 − s)q−1 |Mg ds
Mg
|t1 − t2|q
Γ(q + 1)
Mg
Γ(q)
+
t2
t0 +a
t0
|(t1 − s)q−1 − (t2 − s)q−1| ds
Mg
|t1 − t2|q
Γ(q + 1)
as t1 → t2 ,
(31)
uniformly for all x ∈ X . Hence B(X) is an equi-continuous set in X . Now we apply Arzelá-Ascoli theorem to yield that B(X) is a relatively compact set in X . The
continuity of B follows from the continuity of the function g on J × R.
Finally, since f (t, x) and g(t, x) are nondecreasing in x for each t ∈ J , the operators A and B are nondecreasing on X . Also the hypothesis (H 5 ) yields that u0 Au0 + By for all y ∈ X . Thus, all the conditions of Theorem 3.5 are satisfied and we
conclude that the hybrid fractional integral equation (24) admits a solution. This completes the proof.
R EMARK 4.3. The conclusion of Theorem 4.4 also remains true if we replace
the hypothesis (H 5 ) concerning the existence of lower solution type function by the
following one.
(H 5 ) There exists an element u0 ∈ X = C(J, R) such that
u0 (t) f (t, u0 (t)) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, y(s))ds
H YBRID FIXED POINT THEORY AND APPLICATIONS
179
for all t ∈ J and y ∈ X .
Next, we consider the following nonlinear fractional integral equation of mixed type,
viz.,
t
t
1
x(t) = h(t) + v(t, s) f (s, x(s)) ds +
(t − s)q−1 g(s, x(s)) ds
(32)
Γ(q) t0
t0
for all t ∈ J and 0 < q < 1 , where the functions h : J → R, v : J × J × R and f , g :
J × R → R are continuous.
We consider the following set of hypotheses in what follows.
(H 6 ) The function v : J × J → R+ is continuous. Moreover, V = supt,s∈J |v(t, s)|.
(H 7 ) f (t, x) is linear and nondecreasing in x for each t ∈ J .
(H 8 ) f is bounded on J × R and there exists a constant L > 0 such that | f (t, x)| L|x|
for all t ∈ J and x ∈ R.
(H 9 ) There exists an element u0 ∈ X = C(J, R) such that
u0 (t) h(t) +
t
t0
v(t, s) f (s, u0 (s)) ds +
1
Γ(q)
t
t0
(t − s)q−1 g(s, y(s)) ds
for all t ∈ J and y ∈ X .
R EMARK 4.4. The condition given in hypothesis (H 8 ) is a little more than that of
a lower solution for the HFIE (32) defined on J .
T HEOREM 4.5. Assume that the hypotheses (H 1 )-(H 2 ) and (H 6 ) through (H 9 )
hold. Then the HFIE (32) admits a solution.
Proof. Set X = C(J, R) and define an order relation in X as follows. Let any
x, y ∈ X be arbitrary. Then x y if and only if x(t) y(t) for all t ∈ J . Clearly, as
mentioned earlier, C(J, R) is a lattice. Therefore, the lower and upper bounds for every
pair of elements x, y ∈ X exist. Define the two operators A and B on X by
Ax(t) =
and
Bx(t) = h(t) +
t
t0
v(t, s) f (s, x(s)) ds, t ∈ J,
1
Γ(q)
t
t0
(t − s)q−1 g(s, x(s)) ds, t ∈ J.
(33)
(34)
From the continuity of the integrals involved in the right hand side of (33) and (34) it
follows that A and B define the operators A, B : X → X . We show that A and B satisfy
all the conditions of Theorem 4.5 on X .
180
BAPURAO C. D HAGE
Clearly, the operator A is linear and bounded in view of the hypotheses (H 7 ) and
(H 8 ). We only show that the operator An is partially D -contraction on X for large
positive integer n . Let x, y ∈ X be such that x y. Then by (H 7 )-(H 8 ),
|Ax(t) − Ay(t)| t
t0
V
|V | | f (s, x(s)) − f (s, y(s))| ds
t0 +a
t0
L|x(s) − y(s)| ds
LVax − y.
Taking supremum over t , we obtain
Ax − Ay LVax − y.
Similarly, it can be proved that
A2 x − A2y = |A(Ax(t) − A(Ay(t))|
t0 +a t
|Ax(s) − Ay(s)| ds ds
LV
t0
t0
L2V 2 a2
x − y.
2!
In general, proceeding in the same way, for any positive integer n , we have
LnV n an
x − y.
(35)
n!
for all x, y ∈ X with x y. Therefore, for large n , An is a partially contraction on X .
The rest of proof is similar to that of Theorem 4.4 and now the desired result follows
by an application of Theorem 3.6. This completes the proof.
An x − Any R EMARK 4.5. The conclusion of Theorem 4.5 holds if we replace the condition
(H 9 ) by
(H 9 ) There exists an element u0 ∈ X = C(J, R) such that
u0 (t) h(t) +
t
t0
v(t, s) f (s, u0 (s)) ds +
1
Γ(q)
t
t0
(t − s)q−1 g(s, y(s)) ds
for all t ∈ J and y ∈ X .
The existence theorem for the nonlinear fractional integral equations of mixed type
x(t) = h(t) +
t
t0
v(t, s)x(s) ds +
1
Γ(q)
t
t0
(t − s)q−1 g(s, x(s)) ds, t ∈ J,
(36)
can also be proved by an application of Theorem 3.5 along the lines similar to that of
Theorem 4.5 with appropriate modifications.
We remark that we have applied our hybrid fixed point theorems to initial value
problems of fractional Volterra integral equations, however the idea can be extended to
nonlinear problems of fractional Fredolm integral equations.
H YBRID FIXED POINT THEORY AND APPLICATIONS
181
5. Applications
As the applications of the existence results proved in the previous section, we
consider the initial value problems of nonlinear fractional differential equations.
E XAMPLE 1. Given a closed and bounded interval J = [t0 ,t0 + a] ⊂ R, for some
t0 , a ∈ R with a > 0 and given a real number 0 < q < 1 , consider the IVP of nonlinear
fractional differential equation (in short FDE)
⎫
Dq x(t) = f (t, x(t)), t ∈ J, ⎪
⎬
(37)
⎭
x(t0 ) = x0 = x(t)(t − t0 )1−q ,⎪
t=t0
where Dq is the Riemann-Louville fractional derivative of order q and f : J × R → R
is locally Hölder continuous function.
It is known that the FDE (37) is equivalent to the Volterra fractional integral equation (VFIE)
x(t) = x0 (t − t0 )q−1 +
1
Γ(q)
t
t0
(t − s)q−1 f (s, x(s)) ds, t ∈ J.
(38)
The details of Riemann-Louville fractional derivative and its relations with IVPs may
be found in Lakshmikantham, Leela and Vasundhara Devi [16] and the references cited
therein. The above VFIE (38) is valid for all functions x ∈ C p (J, R). Hence, if hypotheses (H 1 ), (H 2 ) and (H 3 ) hold, then by Theorem 4.1 with h(t) = x0 (t − t0 )q−1 , the FDE
(37) has a solution defined on J .
E XAMPLE 2. Given a closed and bounded interval J = [t0 ,t0 + a] ⊂ R, for some
t0 , a ∈ R with a > 0 and given a real number 0 < q < 1 , consider the IVP of nonlinear
fractional differential equation (in short FDE)
c q
D x(t) = f (t, x(t)), t ∈ J,
(39)
x(t0 ) = x0 ,
where c Dq is the Caputo fractional derivative of order q and f : J × R → R is locally
Hölder continuous function.
It is known that the FDE (39) is equivalent to the Volterra fractional integral equation (VFIE)
t
1
x(t) = x0 +
(t − s)q−1 f (s, x(s)) ds, t ∈ J.
(40)
Γ(q) t0
Again, the details of Caputo fractional derivative and its relations with IVPs may be
found in Lakshmikantham, Leela and Vasundhara Devi [16] and the references cited
therein. The above VFIE (40) is valid for all functions x ∈ C p (J, R). Hence, if hypotheses (H 1 ), (H 2 ) and (H 3 ) hold with h(t) = x0 , then by Theorem 4.1, the FDE (39)
has a solution defined on J .
182
BAPURAO C. D HAGE
E XAMPLE 3. Given a closed and bounded interval J = [t0 ,t0 + a] ⊂ R, for some
t0 , a ∈ R with a > 0 and given a real number 0 < q < 1 , consider the IVP of nonlinear
hybrid fractional differential equation (in short HFDE)
⎫
c q
D [x(t)− f (t, x(t))] = g(t, x(t)), t ∈ J,⎬
(41)
⎭
x0 = x0 ∈ R,
where c Dq is the Caputo fractional derivative of order q , f : J × R → R is continuous
and g : J × R → R is locally Hölder continuous function.
It is easy to verify that the HFDE (41) is equivalent to the hybrid fractional integral
equation (in short HFIE)
x(t) = F(t, x(t)) +
1
Γ(q)
t
t0
(t − s)q−1 g(s, x(s)) ds, t ∈ J,
(42)
where F(t, x(t)) = x0 − f (t0 , x0 ) + f (t, x(t)) for t ∈ J . The details of Caputo fractional
derivatives and its relations with IVPs may be found in Podlubny [19] and the references
cited therein.
Now if x0 − f (t0 , x0 ) 0 and the hypotheses (H 1 ) and (H 5 ) hold, then all the
conditions of Theorem 4.4 are satisfied with the function f replaced by F . Hence, by
Theorem 4.4, the HFDE (41) has a solution defined on J .
E XAMPLE 4. Given a closed and bounded interval J = [t0 ,t0 + a] in R, consider
the following hybrid fractional differential equation with a linear perturbation of second
type,
⎫
Dq [x(t) − f (t, x(t))] = g(t, x(t)), t ∈ J,⎪
⎬
(43)
⎪
⎭
X 0 = X(t)(t − t0 )1−q ,
t=t0
Dq
where
is the Riemann-Louville fractional derivative of order q , 0 < q < 1 , f :
J × R → R is a continuous and g : J × R → R is a locally Hölder continuous function
and X(t) = x(t) − f (t, x(t)), t ∈ J .
The existence theorem for the HFDE (43) is proved using the hybrid fixed point result
embodied in Theorem 3.5 under Lipschitz and compactness type conditions( see Dhage
and Mugale [10]). We mention that the existence theorem for the HFDE (43) can also
be obtained under weaker partially Lipschitz and partially compactness type conditions.
Note that the HFDE (43) is equivalent to the hybrid fractional integral equation (42),
where the function F is defined by
F(t, x(t)) = X 0 (t − t0 )q−1 + f (t, x(t)), t ∈ J.
Therefore, if X0 (t − t0 )q−1 0 and hypotheses (H 1 ) and (H 5 ) hold, then all the conditions of Theorem 4.4 are satisfied with the function f replaced by F . Hence, by
Theorem 4.4, the HFDE (43) has a solution defined on J .
H YBRID FIXED POINT THEORY AND APPLICATIONS
183
R EMARK 5.1. Lastly, we mention that the existence theorems for the hybrid fractional differential equations with the linear perturbation of first kind, namely,
⎫
Dq x(t) = f (t, x(t)) + g(t, x(t)), t ∈ J,⎪
⎬
(44)
⎪
⎭
x0 = x(t)(t − t0 )1−q ,
t=t0
and
c
Dq x(t) = f (t, x(t)) + g(t, x(t)), t ∈ J,
x(t0 ) = x0 ,
(45)
can also be proved along the similar lines with appropriate modifications.
6. Conclusion
In this paper we have proved a very fundamental hybrid fixed point theorems in
partially ordered normed linear spaces. However, more general hybrid fixed point theorems under weaker conditions may be proved along the similar lines with appropriate
modifications. Further these hybrid fixed point theorems have some nice applications
to hybrid differential and integral equations for proving the existence and uniqueness
theorems under weaker hypotheses than earlier ones discussed in the literature. Here,
in this work we have discussed only the dynamic systems with continuous nonlinearity,
however the results on hybrid fixed point theorems of this paper can also be applied to
study the dynamic systems with discontinuous nonlinearities on the domains of their
definition. This can be accomplished by defining the order relation through the appropriate order cone in a normed linear space and following the lines of arguments given in
Hekkillä and Lakshmikantham [13]. Therefore, the work presented in this paper opens
a new vistas for the research work in the area of nonlinear analysis and applications.
Finally, while concluding we mention that some of the results in the stated direction
under weaker hypotheses than that presented here will be reported elsewhere.
Acknowledgements. The author is thankful to the referee for pointing out some
misprints and giving some suggestions for the improvement of this paper.
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(Received December 14, 2012)
Differential Equations & Applications
www.ele-math.com
[email protected]
Bapurao C. Dhage
Kasubai, Gurukul Colony
Ahmedpur-413 515, Dist: Latur
Maharashtra
India
e-mail: [email protected]